Explanation

(d)
$\mathrm{\Delta} = (x + a + b + c)\left| \begin{matrix}
1 & a & b + c \\
1 & b & c + a \\
1 & c & a + b
\end{matrix} \right|\ \ \ \ \ \ \ \ \ \ \ \lbrack C_{1} \rightarrow C_{1} + C_{2} + C_{3}\rbrack$
$= (x + a + b + c)(a + b + c)\left| \begin{matrix}
1 & 1 & b + c \\
1 & 1 & c + b \\
1 & 1 & a + b
\end{matrix} \right|$
$= 0\ \ \ \ \ \ \lbrack C_{2} \rightarrow C_{2} + C_{3}\rbrack$
$Hence,\ x\$may have any value.